Partial Fractions

Suppose we want to find $\int {\frac {3x+1}{x^{2}+x}}dx$ . One way to do this is to simplify the integrand by finding constants $A$ and $B$ so that

${\frac {3x+1}{x^{2}+x}}={\frac {3x+1}{x(x+1)}}={\frac {A}{x}}+{\frac {B}{x+1}}$ .

This can be done by cross multiplying the fraction which gives

${\frac {3x+1}{x(x+1)}}={\frac {A(x+1)+Bx}{x(x+1)}}$

As both sides have the same denominator we must have

$3x+1=A(x+1)+Bx$

This is an equation for $x$ so it must hold whatever value $x$ is. If we put in $x=0$ we get $A=1$ and putting $x=-1$ gives $=-B=-2$ so $B=2$ . So we see that

${\frac {3x+1}{x^{2}+x}}={\frac {1}{x}}+{\frac {2}{x+1}}$

Returning to the original integral

$\int {\frac {3x+1}{x^{2}+x}}dx$	$=\int {\frac {dx}{x}}+\int {\frac {2}{x+1}}dx$
	$=\int {\frac {dx}{x}}+2\int {\frac {dx}{x+1}}$
	$=\ln |x|+2\ln {\Big |}x+1{\Big |}+C$

Rewriting the integrand as a sum of simpler fractions has allowed us to reduce the initial integral to a sum of simpler integrals. In fact this method works to integrate any rational function.

Method of Partial Fractions

To decompose the rational function ${\frac {P(x)}{Q(x)}}$:

• Step 1 Use long division (if necessary) to ensure that the degree of $P(x)$ is less than the degree of $Q(x)$.
• Step 2 Factor Q(x) as far as possible.
• Step 3 Write down the correct form for the partial fraction decomposition (see below) and solve for the constants.

To factor Q(x) we have to write it as a product of linear factors (of the form $ax+b$) and irreducible quadratic factors (of the form $ax^{2}+bx+c$ with $b^{2}-4ac<0$).

Some of the factors could be repeated. For instance if $Q(x)=x^{3}-6x^{2}+9x$ we factor $Q(x)$ as

$Q(x)=x(x^{2}-6x+9)=x(x-3)(x-3)=x(x-3)^{2}$

It is important that in each quadratic factor we have $b^{2}-4ac<0$ , otherwise it is possible to factor that quadratic piece further. For example if $Q(x)=x^{3}-3x^{2}+2x$ then we can write

$Q(x)=x(x^{2}-3x+2)=x(x-1)(x-2)$

We will now show how to write ${\frac {P(x)}{Q(x)}}$ as a sum of terms of the form

${\frac {A}{(ax+b)^{k}}}$ and ${\frac {Ax+B}{(ax^{2}+bx+c)^{k}}}$

Exactly how to do this depends on the factorization of $Q(x)$ and we now give four cases that can occur.

Q(x) is a product of linear factors with no repeats

This means that $Q(x)=(a_{1}x+b_{1})(a_{2}x+b_{2})\cdots (a_{n}x+b_{n})$ where no factor is repeated and no factor is a multiple of another.

For each linear term we write down something of the form ${\frac {A}{(ax+b)}}$ , so in total we write

${\frac {P(x)}{Q(x)}}={\frac {A_{1}}{a_{1}x+b_{1}}}+{\frac {A_{2}}{a_{2}x+b_{2}}}+\cdots +{\frac {A_{n}}{a_{n}x+b_{n}}}$

Example Problem

Find $\int {\frac {1+x^{2}}{(x+3)(x+5)(x+7)}}dx$

Here we have $P(x)=1+x^{2}\ ,\ Q(x)=(x+3)(x+5)(x+7)$ and Q(x) is a product of linear factors. So we write

${\frac {1+x^{2}}{(x+3)(x+5)(x+7)}}={\frac {A}{x+3}}+{\frac {B}{x+5}}+{\frac {C}{x+7}}$

Multiply both sides by the denominator

$1+x^{2}=A(x+5)(x+7)+B(x+3)(x+7)+C(x+3)(x+5)$

Substitute in three values of x to get three equations for the unknown constants,

${\begin{matrix}x=-3&1+3^{2}=2\cdot 4A\\x=-5&1+5^{2}=-2\cdot 2B\\x=-7&1+7^{2}=(-4)\cdot (-2)C\end{matrix}}$

so $A={\tfrac {5}{4}}\ ,\ B=-{\tfrac {13}{2}}\ ,\ C={\tfrac {25}{4}}$ , and

${\frac {1+x^{2}}{(x+3)(x+5)(x+7)}}={\frac {5}{4x+12}}-{\frac {13}{2x+10}}+{\frac {25}{4x+28}}$

We can now integrate the left hand side.

$\int {\frac {1+x^{2}}{(x+3)(x+5)(x+7)}}dx={\tfrac {5}{4}}\ln {\Big |}x+3{\Big |}-{\tfrac {13}{2}}\ln {\Big |}x+5{\Big |}+{\tfrac {25}{4}}\ln {\Big |}x+7{\Big |}+C$

Exercises

Evaluate the following by the method partial fraction decomposition.

1. $\int {\frac {2x+11}{(x+6)(x+5)}}dx$

2. $\int {\frac {7x^{2}-5x+6}{(x-1)(x-3)(x-7)}}dx$

Q(x) is a product of linear factors some of which are repeated

If $(ax+b)$ appears in the factorisation of $Q(x)$ k-times then instead of writing the piece ${\frac {A}{ax+b}}$ we use the more complicated expression

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A_1}{ax+b}+\frac{A_2}{(ax+b)^2}+\frac{A_3}{(ax+b)^3}+\cdots+\frac{A_k}{(ax+b)^k}}

Example 2

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{dx}{(x+1)(x+2)^2}}

Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x)=1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(x)=(x+1)(x+2)^2} We write

${\frac {1}{(x+1)(x+2)^{2}}}={\frac {A}{x+1}}+{\frac {B}{x+2}}+{\frac {C}{(x+2)^{2}}}$

Multiply both sides by the denominator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1=A(x+2)^2+B(x+1)(x+2)+C(x+1)}

Substitute in three values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to get 3 equations for the unknown constants,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} x=0 & 1=2^2A+2B+C \\ x=-1 & 1=A \\ x=-2 & 1=-C \end{matrix}}

so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=1\ ,\ B=-1\ ,\ C=-1} and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{(x+1)(x+2)^2}=\frac{1}{x+1}-\frac{1}{x+2}-\frac{1}{(x+2)^2}}

We can now integrate the left hand side.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{dx}{(x+1)(x+2)^2}=\ln\left|x+1\right|-\ln\left|x+2\right|+\frac{1}{x+2}+C}

We now simplify the fuction with the property of Logarithms.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln\left|x+1\right|-\ln\left|x+2\right|+\frac{1}{x+2}+C=\ln\left|\frac{x+1}{x+2}\right|+\frac{1}{x+2}+C}

Exercise

3. Evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{x^2-x+2}{x(x+2)^2}dx} using the method of partial fractions.

Q(x) contains some quadratic pieces which are not repeated

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2+bx+c} appears we use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{Ax+B}{ax^2+bx+c}} .

Exercises

Evaluate the following using the method of partial fractions.

4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{2}{(x+2)(x^2+3)}dx}

5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{dx}{(x+2)(x^2+2)}}

Q(x) contains some repeated quadratic factors

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2+bx+c} appears k-times then use

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A_1x+B_1}{ax^2+bx+c}+\frac{A_2x+B_2}{(ax^2+bx+c)^2}+\frac{A_3x+B_3}{(ax^2+bx+c)^3}+\cdots+\frac{A_kx+B_k}{(ax^2+bx+c)^k}}

Exercise

Evaluate the following using the method of partial fractions.

6. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{dx}{(x-1)(x^2+1)^2}}

Exercies Solutions

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln\Big|x+6\Big|+\ln\Big|x+5\Big|+C}
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{2}{3}\ln\Big|x-1\Big|-\tfrac{27}{4}\ln\Big|x-3\Big|+\tfrac{157}{12}\ln\Big|x-7\Big|+C}
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\ln\Big|x(x+2)\Big|}{2}+\frac{4}{x+2}+C}
4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln\left(\sqrt[7]{\frac{(x+2)^2}{x^2+3}}\right)+\frac{4\arctan\Big(\tfrac{x}{\sqrt3}\Big)}{7\sqrt3}+C}
5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln\left(\sqrt[12]{\frac{(x+2)^2}{x^2+2}}\right)+\frac{\sqrt2\arctan\Big(\tfrac{x}{\sqrt2}\Big)}{6}+C}
6. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1-x}{4(x^2+1)}+\tfrac{1}{8}\ln\left(\frac{(x-1)^2}{x^2+1}\right)-\frac{\arctan(x)}{2}+C}

Resources

Integration Using Partial Fraction Decomposition Part 1 by James Sousa

Integration Using Partial Fraction Decomposition Part 2 by James Sousa

Partial Fraction Decomposition Part 1 (Linear) by James Sousa

Partial Fraction Decomposition - Part 2 of 2 by James Sousa

Partial Fraction Decomposition - Example 1 by patrickJMT

Partial Fraction Decomposition - Example 2 by patrickJMT

Partial Fraction Decomposition - Example 4 by patrickJMT

Partial Fraction Decomposition - Example 5 by patrickJMT

Partial Fraction Decomposition - Example 6 by patrickJMT

Partial Fraction Decompositions by patrickJMT

Licensing

Content obtained and/or adapted from:

• Partial Fraction Decomposition, Wikibooks: Calculus/Integration techniques under a CC BY-SA license